Polynomial bivariate copulas of degree five: characterization and some particular inequalities

Bivariate polynomial copulas of degree 5 (containing the family of Eyraud-Farlie-Gumbel-Morgenstern copulas) are in a one-to-one correspondence to certain real parameter triplets (a,b,c), i.e., to some set of polynomials in two variables of degree 1: p(x,y) = ax+by+ c. The set of the parameters yielding a copula is characterized and visualized in detail. Polynomial copulas of degree 5 satisfying particular (in)equalities (symmetry, Schur concavity, positive and negative quadrant dependence, ultramodularity) are discussed and characterized. Then it is shown that for polynomial copulas of degree 5 the values of several dependence parameters (including Spearman’s rho, Kendall’s tau, Blomqvist’s beta, and Gini’s gamma) lie in exactly the same intervals as for the Eyraud-Farlie-Gumbel-Morgenstern copulas. Finally we prove that these dependence parameters attain all possible values in ]−1,1[ if polynomial copulas of arbitrary degree are considered.


Introduction
In this paper, we will restrict ourselves to two-dimensional (or bivariate) copulas, and we will simply call them copulas. Recall that a copula is a bivariate cumulative distribution function (restricted to [ , ] ) with uniform marginals on [ , ], which captures the whole dependence structure of the random pair [90].
The name "copula" for functions linking an n-dimensional distribution and its one-dimensional marginals goes back to Sklar [90], in which paper he proved a result which is usually referred to as Sklar's theorem. The exact mathematical relationship for n = is recalled in Theorem 2.1 below. However, links between multivariate distributions and their one-dimensional marginals have been studied before, e.g., by Hoe ding [39,40], Fréchet [36], Dall'Aglio [18][19][20], and Féron [35], and also later on without any reference to the concept of copulas (see, e.g., [81,98]).
There are in nitely many copulas: in the books [43,66] and, more recently, [30] one nds plenty of examples of parametric families (usually with one or two parameters) of copulas and classes of copulas which can be constructed and characterized by functions in one variable (e.g., by additive and/or multiplicative generators [1,58,82,83] in the case of Archimedean copulas).
One of the basic copulas is the independence or product copula Π which is a polynomial of degree two in two variables. As a consequence of, e.g., [92], Π is the only polynomial copula of degree , and there exists no polynomial copula of degree . Polynomial copulas are due to their nature absolutely continuous admitting a representation by their density and discussion of its properties. It is worth noting that quadratic and cubic polynomials have been used in di erent ways to construct (not necessarily polynomial) copulas, e.g., in [11, 22, 51-54, 67, 71-73, 96, 97, 99, 100] (see also [9,69,70]).
The Eyraud-Farlie-Gumbel-Morgenstern copulas (EFGM-copulas) [31,32,38,64] (quite often called Farlie-Gumbel-Morgenstern copulas) form one of the most remarkable families of copulas. It is not di cult to see that these copulas, which are given by (2.5) below and which are widely used when modeling stochastic dependence, are the only polynomial copulas of degree . Unfortunately, all the values of several dependence parameters, as discussed in De nition 6.1, of EFGM-copulas are contained in the interval − , as a consequence of (6.2), so only small dependencies can be modeled. For this reason, several (not necessarily polynomial) extensions and generalizations of EFGM copulas have been considered in the literature (see, e.g., [3,7,8,17,27,41,42,55,56,62,88,89] and the recent survey [78]).
In this contribution, we present a thorough analysis and characterization of polynomial copulas of degree (which are a natural extension of EFGM copulas), and prove some results for the dependence parameters of polynomial copulas of higher degrees.
The paper is organized as follows: after some necessary preliminaries in Section 2, several basic aspects of polynomial copulas of degree are discussed in Section 3. The main results are contained in Section 4 where analytic formulas for the level sets of the parameter set of polynomial copulas of degree are determined with the help of advanced techniques of computer algebra, i.e., by solving a quanti er elimination problem by means of cylindrical algebraic decomposition based on Collins' algorithm [16].
The classi cation of several subclasses of polynomial copulas of degree satisfying additional properties such as symmetry, Schur concavity [80], ultramodularity [60], and positive and negative quadrant dependence [57,66] are given in Section 5. In Section 6, several results concerning dependence parameters as given in De nition 6.1 for polynomial copulas of degree 5 and higher are presented.
Sklar's Theorem [90] (see also [30,44,66]) states that the link between a bivariate probability distribution and its marginals is necessarily a copula. Alternative ways to prove Sklar's Theorem (also for the case of ndimensional probability distributions) can be found in [4,25,26,33,34,68,76]. Stochastically speaking, random variables X and Y whose copula equals Π are called independent, and random variables whose copula equals M or W are called comonotone or countermonotone, respectively. A particularly interesting and important family of copulas is (C EFGM θ ) θ∈[− , ] as de ned below in (2.5), usually referred to as the family of Farlie-Gumbel-Morgenstern copulas [32,38,64,66]. In [30] (see also [13,14,63]) it was pointed out that the corresponding distributions had already been investigated in the earlier and, for many years, forgotten work by Eyraud [31]. In recognition of these early achievements of Eyraud, we will consistently use the name Eyraud-Farlie-Gumbel-Morgenstern copulas (EFGM copulas for short) in this paper. This family of copulas is used quite often when the weak dependence of exchangeable random variables is modeled.
The Eyraud-Farlie-Gumbel-Morgenstern copulas (C EFGM θ ) θ∈ [− , ] have some nice properties: being de ned by polynomials (of degree ), they can be easily computed, and they are absolutely continuous. But the EFGM copulas also have some limitations: under identically distributed marginals they can only generate an exchangeable random pair. Also, the possible values of some well-known dependence parameters are contained in some rather small subintervals of [− , ] (in − , for Spearman's rho, in − , for Kendall's tau, in − , for Blomqvist's beta, and in − , for Gini's gamma, i.e., only weak dependencies can be modeled), and there are linear links between these dependence parameters. We shall have a closer look at dependence parameters of the Eyraud-Farlie-Gumbel-Morgenstern copulas and of polynomial copulas in Section 6.
To keep the advantages and to overcome (some of) the limitations mentioned above, one can consider polynomial copulas of higher degrees (this idea was mentioned but not further developed in [92]). In this paper we shall study copulas de ned by polynomials in two variables and call them polynomial copulas.
For n ∈ N = N ∪ { } consider the polynomial p : R → R in two variables of degree n given by with coe cients a , , a , , a , , . . . , a n, , a n, , . . . , an,n ∈ R. If a n, = a n, = · · · = an,n = then p is called an improper polynomial of degree n, and a proper polynomial of degree n otherwise. For each improper polynomial p of degree n (with the only exception when all coe cients equal -the zero polynomial has no universally accepted degree: sometimes it is said to be unde ned, sometimes it is de ned as deg( ) = − or as deg( ) = −∞) there is a number m ∈ { , , . . . , n − } such that p is a proper polynomial of degree m.
is a copula. Starting with an arbitrary polynomial copula C of degree k, we can use (2.6) to construct a sequence of polynomial copulas (C (n) ) n∈N inductively by Clearly, for each n ∈ N the polynomial copula C (n) is of degree n · k. In particular, if C = Π or C = C EFGM θ with θ ≠ then, for each n ∈ N , the polynomial copula Π (n) constructed via (2.7) is of degree n+ , and the polynomial copula C EFGM θ (n) with θ ≠ is of degree n+ .
(iii) From [59] we know that the function C : [ , ] → [ , ] given by is a polynomial copula of degree which cannot be obtained using construction (2.7), but it can be (compare (2.9) below) rewritten as In [92] it was shown that a copula C : (ii) If p equals the zero polynomial then the corresponding polynomial copula C (of degree ) equals Π, the product copula. (iii) Obviously, there is no proper polynomial copula C with deg(C) = .
(iv) If p is a constant polynomial, i.e., p(x, y) = θ for some θ ∈ R then C is a polynomial copula of degree if and only if θ ∈ [− , ], i.e., C = C EFGM θ .
(v) Generally, if C is a proper polynomial copula of degree n ≥ then the polynomial p has (n− )(n− ) coecients.
It is evident that any polynomial function C : [ , ] → [ , ] which satis es the boundary conditions (C1) of copulas is necessarily of the form (2.9), i.e., For each polynomial p : R → R, this function is absolutely continuous, the mixed derivative ∂ C(x,y) ∂x ∂y exists for each (x, y) ∈ [ , ] , and the density φ C : [ , ] → R is given by (2.10) Then C satis es (C2), i.e., it is -increasing and, therefore, a copula, if and only if φ C (x, y) ≥ for each (x, y) ∈ [ , ] , i.e., C has a non-negative density. Summarizing all the facts which we have mentioned so far, we have obtained the following result: In the monograph [24], where some results from [75] are cited, the following (which has been cited in [65,Subsection 4.4]) is claimed to be an alternative formula for polynomial copulas C : [ , ] → [ , ] of degree n: It is easy to see that formulas (2.11) and (2.9) (as given in [92]) are identical. As we shall see in Example 4.1 below, in general the condition (2.12) is neither su cient nor necessary for C given by (2.11) being a copula. The characterization of the set of polynomials p of degree n ∈ N such that the function given by (2.9) is a (proper) polynomial copula of degree or higher is not trivial at all.
The simplest case is to consider polynomials p : R → R of degree , i.e., p(x, y) = ax + by + c for some a, b, c ∈ R. In this case we have to determine the set of parameters (a, b, c) ∈ R such that C is a copula. In this paper we give a complete characterization and 3D illustrations of the set of parameters (a, b, c) ∈ R such that formula (2.13) yields a copula. In Section 5, possible additional properties of polynomial copulas of degree such as symmetry, Schur concavity, ultramodularity, and both positive and negative quadrant dependence are discussed. Section 6 is devoted to dependence parameters (including Spearman's rho, Kendall's tau, Blomqvist's beta, and Gini's gamma) for polynomial copulas of degree (where they behave in a similar way as for the EFGM copulas) and for polynomial copulas with arbitrary degree (where all possible values in the interval ]− , [ are attained).

Polynomial copulas of degree ve
Our focus is the study of polynomial copulas of degree . It is evident that any polynomial function Then C (a,b,c) satis es (C2), i.e., it is -increasing and, therefore, a copula, if and only if φ C (a,b,c) (x, y) ≥ for each (x, y) ∈ [ , ] , i.e., C (a,b,c) has a non-negative density.
As an immediate consequence of Proposition 2.5 we obtain the following characterization of polynomial copulas of degree .
From [30, Theorem 1.7.5] we know that the limit of each pointwise convergent sequence of copulas is also a copula. In particular, a sequence (C (an ,bn ,cn) ) n∈N of polynomial copulas of degree converges if and only if the three sequences (an) n∈N , (bn) n∈N , and (cn) n∈N converge to some numbers a, b and c, respectively. As a consequence, for a converging sequence (C (an ,bn ,cn) ) n∈N of polynomial copulas of degree we have   The convexity of both the set of all bivariate copulas and the set of all polynomials of degree implies the convexity of the parameter set PC .
It is obvious that for each triplet (a, b, c) ∈ R the assertions "(a, b, c) ∈ PC " and "(b, a, c) ∈ PC " are equivalent. This means that the parameter set PC is symmetric with respect to the plane determined by a − b = , i.e., PC is invariant under re ections through the plane a − b = . We shall denote the intersection of PC with the plane a − b = by PC sym (see Figure 7), i.e., Note that (a, b, c) ∈ PC sym if and only if the corresponding copula C (a,b,c) is symmetric and Schur concave (see Corollary 5.1 and Proposition 5.2). Another consequence is that several dependence parameters (see Section 6) coincide for the copulas C (a,b,c) and C (b,a,c) . For each ζ ∈ R the level set PC [ζ ] of the parameter set PC is de ned by From the convexity of PC it readily follows that each level set PC [ζ ] is also a convex subset of R . In this section we shall give full analytical descriptions of all level sets PC [ζ ] of the parameter set PC of the polynomial copulas of degree . These formulas were obtained with the help of the computer algebra software Mathematica ® .
Note rst that we can rewrite the density φ C (a,b,c) in a way that the inequality (3.2) looks as follows: We see immediately that the summands of the density φ C (a,b,c) change their sign whenever x, y ∈ { , }, and that the monotonicity of the density changes whenever x = or y = .
For the remaining points in , , , , \ ( , ) we obtain necessary conditions for a nonnegative density φ C (a,b,c) which are given in Figure 1.
Obviously, the inequalities in Figure 1 are highly redundant, and we can reduce them to the following equivalent system of necessary conditions: Using, for an arbitrary subset A of R n , the notation Conv(A) for the convex hull of A, i.e., the set of all convex combinations of nitely many points in A given by An equivalent formulation is that a triplet (a, b, c) ∈ PC necessarily has to be a solution of the system of linear inequalities This means that the conditions (4.6) are only necessary but not su cient, and that the subset relation in (4.7) must be strict. As can be seen from Figure 4, P (B ,B ,B ,B ) is the smallest polyhedron containing PC as a subset.

Example 4.1.
Recall the formula of a polynomial copula of degree n given by (2.11) subject to condition (2.12).

. Quanti er Elimination
Recall that the task of identifying all polynomial copulas of degree consists in determining all real numbers a, b, c such that the inequality (3.2) holds for all (x, y) ∈ [ , ] , subject to the linear constraints (4.6). In terms of formal logic, these requirements can be phrased as a quanti er elimination problem: we have a formula Φ consisting of polynomial equations and inequalities in the variables x, y, a, b, c, and we are interested in a formula Ψ consisting of polynomial equations and inequalities in the variables a, b, c such that Ψ is equivalent to ∀x, y : Φ as a statement about the real numbers. Such quanti er elimination problems can be solved using computer algebra. It was rst shown by A. Tarski [95] that the problem is decidable, but the algorithm he proposed is impractical and only of theoretical interest. G. E. Collins [16] has later proposed a more e cient (though still computationally very expensive) algorithm, which is implemented in Mathematica and elsewhere [12,85,94]. This technique is extremely powerful and deserves to be better known.
In principle, our quanti er elimination problem can be solved by the following single Mathematica command: It would even be possible to leave out the linear constraints (4.6), the solution set will be PC in either case. Unfortunately however, in neither of the two variants does the computation come to an end within a reasonable amount of time. In order to solve the problem, we need to employ a more low-level command. In order to explain our computation, we rst need to give a bit more detail on Collins' algorithm. For further details, see [6,15,45].
Collins' algorithm takes as input a system of polynomial equations and/or inequalities in one or more variables with rational coe cients and returns a logical formula with a very particular structure that describes the solution set of the input system.
For example, applied to the single input inequality x + y − ≤ , the algorithm as implemented in the Mathematica command CylindricalDecomposition [93] will produce something like this: This looks more complicated at rst glance, but the structure of the formula has the useful feature that it makes quanti er elimination simple. For example, if we want to know all real numbers x such that ∃y : x + y − ≤ , all we need to do is to replace all the equations and inequalities in the above formula involving y by True and simplify the result. This will give The structure of the output formula can also be used to deal with formulas involving ∀, but it is slightly more di cult to explain precisely how it works, and it is not necessary to do so because we can always use ∀y : Φ ⇐⇒ ¬∃y : ¬Φ to reduce the task to an existential quanti er.
Almost the entire computation time of Collins' algorithm is spent on determining truth values of formulas in areas of the solution set which are speci ed by equations that were introduced during the computation. For this reason, the alternative command GenericCylindricalDecomposition is provided by Mathematica.
It performs Collins' algorithm without paying attention to what happens for the solutions of these equations. For example, applied to x + y − ≥ , this command produces the pair as output. This is to be read as follows: "I have not investigated any points (x, y) ∈ R with x − = or x + = or − + x + y = , but for all other points, I guarantee that x + y − ≥ is true if and only if The rst part of the output again has a format that makes quanti er elimination simple, but the e ect of ignoring a lower-dimensional algebraic set has to be properly taken into account. In the example above, we are entitled to conclude that for almost all real numbers x, the formula x < − ∨ x > is equivalent to the formula ∀ ∼ y : x + y − ≥ , where ∀ ∼ also means "for almost all".
Let us now return to the polynomial inequality (3.2) and the linear constraints (4.6). While we were not able to solve the quanti er elimination problem using Resolve or CylindricalDecomposition, it turns out that GenericCylindricalDecomposition does the job. Note that this is su cient for our purpose, because PC is a convex and, as a consequence of (3.3) and (4.7), also a compact subset of R . Therefore, any solution on some hypersurface passing through the solution set can be seen as the limit of a sequence and/or the convex combination of parameter tuples ful lling the conditions of the guaranteed solution part of the algorithm. Moreover, due to the continuity of copulas there is no harm in using the quanti er ∀ ∼ instead of ∀ on x and y.
Since we prefer to eliminate existential quanti ers, we bring the problem ∀x, y : y, a, b, c). Here, Ψ contains the requirement that x, y are restricted to [ , ] and a, b, c satisfy the inequalities (4.6), and Ξ is (3.2). Applying the command terminates within a few minutes. Depending on the order in which the variables a, b, c are speci ed, the computation time on a Linux machine with 3GHz CPU and 700Gb RAM ranges between 98 and 251 seconds. The computation also terminates when the inequalities of (4.6) are not taken into account, but the runtime then ranges between 4343 and 14856 seconds. In both variants, the output is extremely large, so that it was impossible to work out the quanti er elimination step by hand, but it is easy to let Mathematica replace all inequalities involving x or y by True. The instructions we used to this end are given in the appendix. After quanti er elimination, the formula is smaller, but still very large. One reason is that the formula produced by GenericCylindricalDecomposition uses False as truth value for all points on the disregarded hypersurface. As a consequence, it contains many subformulas of the form u < a < v ∨ v < a < w which, due to the compactness and convexity of the solution set, we can safely simplify to u < a < w. The code we used for this simpli cation is also given in the appendix.
At this point, we have a formula Ψ in a, b, c which is equivalent (possibly up to a set of measure zero) to ∃ ∼ x, y : Ψ(x, y, a, b, c) ∧ ¬Ξ(x, y, a, b, c), where ∃ ∼ means that the set of points (x, y) ∈ R with the required property has positive measure. It remains to perform the negation. For getting a more readable result, we call GenericCylindricalDecomposition on the conjunction of (4.6) and ¬Ψ and apply our simplifying routine to the output.

. Results
We now present the result of the computations described above.
For each c ∈ R the level set PC [c] of the parameter set PC (which is necessarily a convex set) is fully described by giving the following information for the triplets (a, b, c) ∈ PC [c] : (i) the range of c in each of the ve cases to be considered,  (ii) the corresponding range of a (given c), (iii) the corresponding lower and the upper bound for b (given c and a). Table 1 gives a survey of the ve cases which have to be distinguished for the values c and, in Table 2, the complete formulas for the lower and upper bounds of b (given c and a) in each of these ve cases are listed.
To be precise, we summarize in each of the cases (I)-(V) the respective ranges of the level c and of the value a (given c). In (4.12)-(4.16) in Table 2 we give the complete formulas for the lower bounds lb [c] (N) (a) and the upper bounds ub [c] (N) (a) of the value b (given (c, a)) such that (a, b, c) ∈ PC [c] . When looking at the Cases (I)-(V) in Table 1 we immediately see that there is a minimal and a maximal value of c such that there exists a parameter triplet (a, b, c) ∈ PC : The level set PC [ζ * ] = PC [− ] is visualized in Figure 4 (the darker set on the right-hand side), while the level set PC [ζ * ] turns out to be a singleton: Adam Šeliga et al.
Numerical integration tells us that the volume of the parameter set PC has the approximate value of .
cubic units. Since the volume of the pyramid P (B ,B ,B ,B ) (4.5) equals cubic units, the volume of PC corresponds to approximately . % of the volume of P (B ,B ,B ,B ) . The distribution of the areas of the level sets PC [ζ ] of PC on the interval − , + / (in comparison to the areas of the level sets of the polyhedron P (B ,B ,B ,B ) ) is visualized in Figure 6.
Apparently, the projection of the parameter set PC to the ab-plane is not only symmetric with respect to the main diagonal a − b = (which follows from (2.13) by construction). It seems (see Figure 5) that this projection of PC is also symmetric with respect to the opposite diagonal a+b = and to the lines determined by a = and b = . Taking into account the result of [92] and the structure of the formula for C (a,b,c) in (2.13), we obtain the following obvious result:  The set PC sym as given in (4.17) is visualized in Figure 7. Note that each (a, a, c) ∈ PC sym can also be characterized as follows:

Polynomial copulas of degree ve satisfying some particular (in)equalities . Symmetric polynomial copulas of degree ve
if c ∈ , + .

. Schur concave polynomial copulas of degree ve
Schur convexity and Schur concavity (as its dual) were introduced in [80] as variants of the convexity of real functions (see also [74]). Schur convex functions preserve a particular preorder called majorization [61] and play a role in some related inequalities [87].
we have In particular, a copula C : Clearly, the three basic copulas W, Π and M are Schur concave, as well as each associative copula (see [29,47]). It turns out that the set of Schur concave polynomial copulas of degree coincides with the set of symmetric polynomial copulas of degree : Proof. Since each Schur concave copula is symmetric [29], each Schur concave polynomial copula C (a,b,c) satis es (a, b, c) ∈ PC sym, i.e., a = b, which means that (i) implies (ii).
In order to show that also (ii) implies (i) we have to consider points (x, y) ∈ [ , ] with a xed sum u = x + y, i.e., u ∈ [ , ]. Observe that u = and u = are equivalent to (x, y) = ( , ) and (x, y) = ( , ), respectively, so we may restrict ourselves to u ∈ ] , [ because the Schur concavity on a single point is trivial. Since in this part of the proof our copulas are assumed to be symmetric we may also restrict ourselves , u → R given by C(·, u − ·) (x) = C(x, u − x) is strictly increasing. For a symmetric polynomial copula C (a,a,c) of degree , i.e., with (a, a, c) ∈ PC sym, and for each u ∈ ] , [ this means that the derivative of this section of C (a,a,c) with respect to x is non-negative or, equivalently and in a formal way, for each (u, x) ∈ ] , [ × max(u − , ), u we have

. Ultramodular polynomial copulas of degree ve
If A is a non-empty subset of R n then a function f : A → R n is called ultramodular [60] if its increments are non-decreasing. This means that for all u, v ∈ A with u ≤ v and for all h ∈ R n such that In the special case n = , a copula C : [ , ] → [ , ] is ultramodular (see [47][48][49]77]) if and only if for all (u , u ), (v , v ), (w , w ) ∈ [ , ] satisfying (u , u ) + (v , v ) + (w , w ) ∈ [ , ] the following inequality holds: In a geometrical formulation, an ultramodular copula C is characterized by the fact that all onedimensional vertical and horizontal sections are convex (see [ The concepts of ultramodular copulas and of stochastically decreasing copulas [5,57,66,86] are equivalent (see Subsection 5.6 for some details). If we insert into (5.5) the four points in (x, y) ∈ { , } we obtain the following system of linear inequalities which has to be satis ed by any triplet (a, b, c) ∈ PC corresponding to an ultramodular polynomial copula of degree : Since (a, b, c) ∈ P (B ,B ,B ,B ) because of (4.7), (a, b, c) must also solve the system of linear inequalities (4.6).
Using the Mathematica command Reduce, we see that the set of solutions of the joint systems (5.6), (5.7) and (4.6) of linear inequalities coincides with the polyhedron P UM (5.4), i.e., (a, b, c) ∈ P UM . Since for each vertex (a * , b * , c * ) of P UM the density φ C (a * ,b * ,c * ) is easily seen to be non-negative (i.e., satis es (3.2) for each (x, y) ∈ [ , ] ), we have (a * , b * , c * ) ∈ PC and, because of the convexity of PC , also P UM ⊆ PC . Conversely, if for the polynomial copula C (a,b,c) we have (a, b, c) ∈ P UM , then (a, b, c) solves the joint systems (5.6) and (5.7) of linear inequalities and, as a consequence, all horizontal and vertical one-dimensional sections of C (a,b,c) are convex, i.e., C (a,b,c) is ultramodular.
Obviously, the vertex ( , , ) of the convex set P UM corresponds to the product copula Π which is the greatest ultramodular copula.

. Positive quadrant dependence
If (Ω, A, P) is a probability space, X, Y : Ω → R are continuous random variables and C : [ , ] → [ , ] is the corresponding copula according to Theorem 2.1, then X and Y (and C) are said to be positively quadrant dependent (PQD) [57,66] if, for all (u, v) ∈ R (5.8) or (recalling the product copula Π = C ( , , ) ) if C ≥ Π. Proof. Let us rst determine which polynomial functions C (a,b,c) as given in (2.13) satisfy the two conditions C (a,b,c) ≥ Π and (a, b, c) ∈ P (B ,B ,B ,B ) .
To verify C (a,b,c) ≥ Π it su ces, because of the convexity of [ , ] and the convexity of the function (x, y) −→ ax + by + c, to check the non-negativity of ax + by + c for each vertex (x, y) ∈ { , } of the unit square, which exactly means that the triplet (a, b, c) satis es the following system of linear inequalities: We already know that (a, b, c) ∈ P (B ,B ,B ,B ) if and only if (a, b, c) solves the system (4.6) of linear inequalities.
Subtracting the fourth inequality from the sum of the rst two inequalities in (4.6), the joint system of the inequalities (5.10) and (4.6) can be rewritten as It is readily seen (e.g., using again a Mathematica command like Reduce) that (a, b, c) solves (5.11) if and only if (a, b, c) ∈ P PQD as given by (5.9). This means that (a, b, c) ∈ P PQD if and only if a polynomial function C (a,b,c) satis es C (a,b,c) ≥ Π and (a, b, c) ∈ P (B ,B ,B ,B ) .
Since  triplet (a, b, c) is an element of the subset P NQD of PC given by For each c ∈ R the level set P NQD [c] of the set of parameters P NQD of the set of all negatively quadrant dependent polynomial copulas of degree is characterized as follows: The parameter sets P PQD and P NQD of positively and negatively quadrant dependent polynomial copulas of degree , respectively, are visualized in Figure 9. Obviously, we have P PQD ∩ P NQD = {( , , )}, i.e., the product copula Π = C ( , , ) is the only (polynomial) copula (of degree ) which is both negatively and positively quadrant dependent, as expected. Since each ultramodular copula is necessarily negatively quadrant dependent, we also have P UM ⊆ P NQD .

. Some automorphisms of polynomial copulas of degree ve
If C denotes the set of (bivariate) copulas, the mappings Obviously, each of the involutive bijections induced by (5.15) transforms a polynomial copula C (of arbitrary degree n) into a polynomial copula C of the same degree, i.e., the mappings in (5.15) induce also involutive bijections from PC onto PC .
Since each polynomial copula C of degree can be identi ed with a parameter triplet (a, b, c) ∈ PC , i.e., C = C (a,b,c) as de ned in (2.13), we obtain the following formulas for the x-ipping, the y-ipping and the survival copula of a polynomial copula of degree : showing that the three mappings C −→ C x ip , C −→ C y ip and C −→ C given in (5.15) induce involutive bijections from PC onto PC . From (5.16) it follows immediately which polynomial copulas of degree are invariant under these automorphisms.
Corollary 5.6. Let C (a,b,c) : [ , ] → [ , ] be a polynomial copula of degree , i. e., (a, b, c copula C (a,b,c) is invariant under x-ipping, i.e., (C (a,b,c) C (a,b,c) , if and only if there is a parameter θ ∈ [− , ] such that (a, b, c) = (− θ, , θ). copula C (a,b,c) is invariant under y-ipping, i.e., (C (a,b,c) ) y ip = C (a,b,c Since the Eyraud-Farlie-Gumbel-Morgenstern copulas are polynomial copulas of degree , Corollary 5.6 (iii) tells us that no proper polynomial copula of degree 5 coincides with its survival copula. Therefore, polynomial copulas of degree are more exible than EFGM copulas since they do not induce radial symmetry. Note also that, for bivariate copulas, the re ections considered in [28] (see also [50]) are exactly the xippings (5.12) and y-ippings (5.13) (which is no longer the case if one also considers n-dimensional copulas with n > ).
It is easy to see that the x-ipping and the y-ipping of a copula C change its relationship to the product copula Π. Therefore, for each (a, b, c) ∈ PC we have (recall that Π x ip = Π y ip = Π) As a consequence, the functions C −→ C x ip and C −→ C y ip in (5.15) can be seen as bijections between the sets of positively quadrant dependent, on the one hand, and negatively quadrant dependent, on the other hand, polynomial copulas of degree , while the construction of the survival copula C −→ C in (5.15) acts as an involution on both positively and negatively quadrant dependent polynomial copulas of degree : (iv) The survival copula (C (a,b,c) ) of the copula C (a,b,c) is positively quadrant dependent.
By duality, we obtain another set of four equivalent assertions if we replace in Corollary 5.7 consistently the property "positively quadrant dependent" by "negatively quadrant dependent" and vice versa.
We only mention that the x-ipping, the y-ipping and the construction of the survival copula of a polynomial copula C (a,b,c) of degree map each extremal point of any of the convex parameter sets P PQD and P NQD to an extremal point of the respective range.
As already mentioned in Section 5.3, there is a special relationship between ultramodular copulas and stochastically decreasing copulas (see [66] (iii) C x ip is stochastically increasing.
(iv) Each horizontal and each vertical section of C x ip is concave.
(v) C y ip is stochastically increasing.
(vi) Each horizontal and each vertical section of C y ip is concave.
Finally Turning our attention to polynomial copulas of degree , we are able to identify the parameter sets of stochastically decreasing and increasing polynomial copulas of degree .

Dependence parameters of polynomial copulas of degree ve and higher
Given two random variables X and Y, the full information about their dependence is contained in the corresponding copula (see Theorem 2.1). A number of dependence parameters (also called concordance measures) measure the degree of dependence of X and Y (or, equivalently, of C). In this paper we shall deal with Spearman's rho [91], Kendall's tau [46], Blomqvist's beta [10], and Gini's gamma [37].
De nition 6.1. Let (Ω, A, P) be a probability space and let X, Y : Ω → R be continuous random variables with copula C : [ , ] → [ , ]. The following dependence parameters are de ned by ϱ X,Y = ϱ C = [ , ] C(x, y) dx dy − , (Spearman's rho) The Fréchet-Hoe ding lower bound W, the product copula Π and the Fréchet-Hoe ding upper bound M given by (2.4) are characterized by the following properties of the dependence parameters considered in De nition 6.1: For the family of Eyraud-Farlie-Gumbel-Morgenstern copulas (C EFGM θ ) θ∈[− , ] the corresponding dependence parameters (for more details see [30,44,66]) are given as follows: As a consequence, for each θ ∈ [− , ] we have If we switch to polynomial copulas of degree , it is easy to show that the dependence parameters of C (a,b,c) can be computed in a simple way as functions of the parameters a, b and c: Corollary 6.2. For each polynomial copula of degree , i.e., for each triplet (a, b, c) ∈ PC , we have (i) ϱ C (a,b,c) = (a + b + c); It is quite interesting that the values of the four dependence parameters considered here cover the same intervals for polynomial copulas of degree as for polynomial copulas of degree , i.e., for EFGM copulas. Moreover, because of the symmetry of the parameter set PC we obtain the same intervals if we only consider the symmetric polynomial copulas of degree C (a,a,c) , i.e., with (a, a, c) ∈ PC sym. Proposition 6.3. For polynomial copulas of degree the following equalities hold: Proof. Note rst that as a consequence of (4.6) and the fact that each EFGM copula C EFGM θ = C ( , ,θ) with θ ∈ [− , ] is a(n improper) polynomial copula of degree . Therefore, the validity of (i), (iii) and (iv) follows immediately from Corollary 6.2 (i),(iii),(iv).
In other words, this means that the minimum of The results in this section allow us to formulate some preliminary observations concerning the so-called τ-ϱ-region of the set of polynomial copulas of degree which describes the relationship between the dependence parameters τ and ϱ. Recall that for a set S ⊆ C of copulas the τ-ϱ-region R [τ,ϱ]  [66], and compare also [79]).
Clearly, for the set of EFGM copulas we obtain a line segment: From Corollary 6.2 (i)-(ii) and Proposition 6.3 (i)-(ii) it follows that the τ-ϱ-region R [τ,ϱ] PC of all polynomial copulas of degree is a proper subset of the rectangle − , × − , (taking into account [66, Figure 5 given by (2.6) is a copula (see [53,Theorem 1] and Example 2.3 (ii)). Evidently, if C is a polynomial copula of degree k then D C is a polynomial copula of degree k. De ne the sequence of copulas (E−n) n∈N in C [poly] inductively by Then, following [47,52], the sequence (E−n) n∈N converges to the Fréchet-Hoe ding lower bound W, i.e., lim n→∞ E−n = W . (6.3) Fix an arbitrary r ∈ R \ Z with r < , note that there is a unique nr ∈ N such that −nr − < r < −nr, and put Er = −(r + nr)E −(nr+ ) + (r + nr + )E−n r .
We see that Er is a convex combination of the two copulas E−n r and E −(nr+ ) , and thus also a copula. Then the parametric family (Er) r∈]−∞, ] of copulas is continuous and monotone non-decreasing with respect to the parameter r. Taking into account (6.3) and the fact that W and Π are the only copulas where all the dependence parameters given in De nition 6.1 are equal to − and , respectively (see (6.1)), we obtain To conclude the proof, it is enough to consider the x-ipping C x ip : [ , ] → [ , ] of a copula C [21] given by (5.12) which is also a copula. Note that for each copula C we have ϱ C x ip = −ϱ C , τ C x ip = −τ C , β C x ip = −β C , and γ C x ip = −γ C . Moreover, if C is a polynomial copula (of degree k) then also C x ip is a polynomial copula (of degree k). Since each copula Er is symmetric we even have shown the validity of the result of Proposition 6.4 if we restrict ourselves to symmetric copulas, i.e., to parameters in (a, a, c) ∈ PC sym:

Concluding remarks
Polynomial copulas can be seen as a genuine generalization of Eyraud-Farlie-Gumbel-Morgenstern copulas (which are, in fact, just polynomial copulas of degree ). We have focused on polynomial copulas of degree which necessarily have the form (2.13), i.e., C (a,b,c) (x, y) = xy + (ax + by + c)x( − x)y( − y) for suitable parameters (a, b, c) ∈ R . This parameter set PC ⊂ R was fully characterized and illustrated, giving analytical descriptions by means of inequalities of all level sets PC [ζ ] = {(a, b, c) ∈ PC | c = ζ }, i.e., for each ζ ∈ − , + / (see Tables 1-2). Similarly, we have considered and characterized additional properties of polynomial copulas of degree such as symmetry, Schur concavity, ultramodularity, and positive and negative quadrant dependence.
For each triplet (a, b, c) ∈ PC several dependence parameters (Spearman's rho, Kendall's tau, Blomqvist's beta and Gini's gamma) with respect to the corresponding polynomial copulas were computed. As can be seen from Corollary 6.2, here the expression a + b + c plays a crucial role. As a consequence of (4.6), in PC this term a + b + c attains all values in the interval [− , ], and it assumes the extremal values − and for all parameters (a, −a, − ) with a ∈ [− , ] and ( − c, − c, c) with c ∈ [ , ], respectively. An interesting result is that Spearman's rho, Kendall's tau, Blomqvist's beta and Gini's gamma cover the same set of values for polynomial copulas of degree as for polynomial copulas of degree , i.e., for EFGM copulas (Proposition 6.3). The situation changes if we consider polynomial copulas of arbitrary degree, in which case these four dependence parameters cover the maximal interval ]− , [ (Proposition 6.4 and Corollary 6.5).
For polynomial copulas of higher degrees, the number of coe cients to be taken into account grows quadratically. For proper polynomial copulas of degree n ≥ we have to consider (n− )(n− ) coe cients. For example, if n = we have to deal with sixtuples (a , a , . . . , a ) ∈ R , and all polynomial copulas of degree necessarily have the form C(x, y) = xy + (a x + a xy + a y + a x + a y + a )x( − x)y( − y). (7.1) As an immediate consequence of Proposition 2.5 we must have ∂ C(x,y) ∂x ∂y ≥ which means that, e.g., for (x, y) = ( , ) we obtain a ≥ − (compare this with the corresponding inequality c ≥ − for polynomial copulas C (a,b,c) of degree 5 in Figure 1).
As mentioned in Example 2.3 (iii) (see [59]) is a polynomial copula of degree (compare the results of [92] and (2.9)). It therefore corresponds to the -tuple , , , − , − , − . However, no characterization of all -tuples (a , a , . . . , a ) ∈ R yielding a copula as given by (7.1) is known so far. 4. De ne a function that performs the whole computation explained in Subsection 4.1, for a particular given variable order. The input argument should be a list of ve variables such that the rst three variables are a, b, c (in any order) and the last two are x, y (in any order). For any such input, the function returns a formula equivalent to the formula described in Subsection 4.2.